A vector field on Ω is a function F : Ω → T Ω such that F(a

1. Vector Analysis v.s. Differential Forms

Let Ω be an open subset of R³. A vector field on Ω is a function F : Ω → T Ω such that F(a) ∈ Ta(R³). For each a ∈ Ω, we write

F(a) = P (a)(e₁)_a+ Q(a)(e₂)_a+ R(a)(e₃)_a∈ T_a(R³).

We obtain functions P, Q, R : Ω → R. We say that F is a smooth vector field on Ω if P, Q, R are smooth.

Example 1.1. If f : Ω → R is a smooth function, the gradient grad f of f is a vector field on Ω :

grad f : Ω → T Ω defined by

(grad f )(a) = f_x(a)(e₁)_a+ f_y(a)(e₂)_a+ f_z(a)(e₃)_a. If F is a smooth vector field on Ω, we define the curl of F by

(curl F )(a) = ((∇ × F)(a))a, where

∇ × F =      

e1 e2 e3

∂_x ∂_y ∂_z

P Q R

= (R_y− Q_z, P_z− R_x, Q_x− P_y).

Hence curl F defines a vector field on Ω and is given by

curl F(a) = (Ry(a) − Qz(a))(e1)a+ (Pz(a) − Rx(a))(e2)a+ (Qx(a) − Py(a))(e3)a. If F is a vector field on Ω, we define the divergence of F by

(div F)(a) = P_x(a) + Q_y(a) + R_z(a).

Then div F : Ω → R is a smooth function.

We associate to any smooth vector field F on Ω a one form ω_F on Ω and a two form η_F on Ω defined by

ω_F= P dx + Qdy + Rdz

ηF= P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy.

By definition,

df = fxdx + fydy + fzdz = ωgradf. We also see that

dω_F= (P_xdx + P_ydy + P_zdz) ∧ dx + (Qxdx + Qydy + Qzdz) ∧ dy + (Rxdx + Rydy + Rzdz) ∧ dz

= (R_y − Q_z)dy ∧ dz + (P_z− R_x)dz ∧ dx + (Q_x− P_y)dx ∧ dy

= ηcurl F.

If c is a 2-chain, Stoke’s Theorem implies that Z

∂c

ωF = Z

c

dωF.

1

2

Since dωF = ηcurlF, we obtain the Stoke’s Theorem in vector analysis:

Z

∂c

P dx + Qdy + Rdz = Z

c

(R_y− Q_z)dy ∧ dz + (P_z− R_x)dz ∧ dx + (Q_x− P_y)dx ∧ dy.

We can also compute dηF:

dη_F= (Pxdx + Pydy + Pzdz) ∧ dy ∧ dz + (Q_xdx + Q_ydy + Q_zdz) ∧ dz ∧ dx + (R_xdx + R_ydy + R_zdz) ∧ dx ∧ dy

= (div F)dx ∧ dy ∧ dz If c is a 3-chain, Stoke’s Theorem implies that

Z

∂c

η_F = Z

c

dη_F.

Since dηF = (div F)dx ∧ dy ∧ dz, we obtain the Gauss divergence theorem in vector analysis:

Z

∂c

P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy = Z

c

(div F)dx ∧ dy ∧ dz.

Furthermore, using d²= 0, we can obtain the following results. We obtain that d(dωF) = dηcurl F= (div curl F)dx ∧ dy ∧ dz.

Since d(dωF) = 0, we find that

div curl F = 0.

Since d(df ) = d(ωgrad f) = ηcurl gradf, and d²f = 0, we find that curl grad f = 0.

Proposition 1.1. Let F, G be vector fields on an open subset Ω of R³ and f : Ω → R be a smooth function.

(1) F = grad f if and only if ωF = df.

(2) curl F = 0 if and only if dω_F = 0.

(3) F = curl G if and only if ηF= dωG. (4) div F = 0 if and only if dη_F = 0.

Proposition 1.1 allows us to study the following important problems in vector analysis.

Corollary 1.1. Let B be any open ball in R³ and F be a vector field on B. Then we have the following properties:

(1) div F = 0 if and only if there exists a vector field G on B so that F = curl G.

(2) curl F = 0 if and only if there exists a smooth function f : B → R so that F = grad f.

In physics, f is called a potential function for the vector field F.

Proof. The proof of this Corollary follows from the Poincare Lemma and Proposition 1.1.